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Thread: Gaussian Integral

  1. #1
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    Gaussian Integral

    Heya having a complete mind blank today can anyone give me a hint on how to get started with this
    $\displaystyle
    \int_0^\infty x^2e^{-x^2}dx
    $

    Cheers
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  3. #3
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    Let $\displaystyle u=x^{2}, \;\ \frac{1}{2}du=xdx$

    Make the subs and we get a form we can easily use the gamma function.

    We get upon making the subs:

    $\displaystyle \frac{1}{2}\int_{0}^{\infty}u^{\frac{1}{2}}e^{-u}du$

    Now, use $\displaystyle {\Gamma}(p)=\int_{0}^{\infty}u^{p-1}e^{-u}du$

    See what p must be?. p-1=1/2
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    Hello,

    $\displaystyle \int_0^\infty x^2 e^{-x^2} ~ dx=-\frac 12 \int x(-2x)e^{-x^2} ~ dx$
    Integrate by parts :
    $\displaystyle =-\frac 12 \left[xe^{-x^2}\right]_0^\infty+\frac 12 \int_0^\infty e^{-x^2} ~ dx=\frac 12 \int_0^\infty e^{-x^2} ~ dx$

    Knowing that $\displaystyle \int_0^\infty e^{-x^2} ~ dx=\frac{\sqrt{\pi}}{2}$, we then have :

    $\displaystyle I=\frac{\sqrt{\pi}}{4}$
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